# Math Help - Boolean ring

1. ## Boolean ring

Prove that in a Boole ring with elements there exists s.t. for all distinct .

2. Originally Posted by petter
Prove that in a Boolean ring with elements there exists s.t. for all distinct .
since the ring $R$ is Boolean, it is unitary, commutative and has no non-zero nilpotent element. hence $R$ is the direct sum of $n$ copies of $\mathbb{F}_2.$ now for any $1 \leq i \leq n$ let $a_i=(x_{i1}, \cdots , x_{in}) \in R,$

where $x_{ii}=1$ and $x_{ik}=0, \ \forall k \neq i.$ obviously $a_ia_j=0, \ \forall i \neq j. \ \ \Box$